CapacitanceCapacitance

AP Physics BAP Physics B

CapacitorsCapacitorsConsider two separated conductors, like two Consider two separated conductors, like two parallel plates, with external leads to attach to parallel plates, with external leads to attach to other circuit elements. Such a device is called other circuit elements. Such a device is called a a capacitorcapacitor. .

There is a limit to the amount of charge that a conductor can hold without leaking to the air. There is a certain capacity for holding charge.

There is a limit to the amount of charge that a conductor can hold without leaking to the air. There is a certain capacity for holding charge.

CapacitanceCapacitance

The capacitance (C) of a conductor is defined as the ratio of the charge (Q) on the conductor to the potential (V) produced.

The capacitance (C) of a conductor is defined as the ratio of the charge (Q) on the conductor to the potential (V) produced.

Capacitance:

; : Q

C Units Coulombs per voltV

; : Q

C Units Coulombs per voltV

Capacitance in FaradsCapacitance in Farads

One One farad (F)farad (F) is the capacitance is the capacitance CC of a conductor that of a conductor that holds one coulomb of charge for each volt of potential.holds one coulomb of charge for each volt of potential.

(C); (F)

(V)

Q coulombC farad

V volt

(C); (F)

(V)

Q coulombC farad

V volt

Example:Example: When 40 When 40 C of charge are placed on a C of charge are placed on a con- ductor, the potential is 8 V. What is the con- ductor, the potential is 8 V. What is the capacitance?capacitance?

40 C

8 V

QC

V

C = 5 FC = 5 F

Parallel Plate CapacitanceParallel Plate Capacitance

d

Area A+Q

-Q

You will recall from Gauss’ law that You will recall from Gauss’ law that EE is is also:also:

and Q V

C EV d

For these two parallel plates:

0 0

QE

A

0 0

QE

A

QQ is charge on either is charge on either plate. plate. AA is area of is area of plate.plate.

0

V QE

d A AndAnd

0

Q AC

V d 0

Q AC

V d

Example 2.Example 2. The plates of a parallel The plates of a parallel plate capacitor have an area of plate capacitor have an area of 0.4 m0.4 m22 and are 3 mm apart in air. and are 3 mm apart in air. What is the capacitance?What is the capacitance?

3 mmd

A

0.4 m2

0

Q AC

V d 0

Q AC

V d

2

2

-12 2CNm

(8.85 x 10 )(0.4 m )

(0.003 m)C

C = 1.18 nFC = 1.18 nF

Applications of CapacitorsApplications of Capacitors

+++++++

-

-

---

-- A

Variable Capacitor

Changing Area

0

AC

d 0

AC

d

d

Changing d

Microphone

QV

C

QV

C

A A microphonemicrophone converts sound waves into an converts sound waves into an electrical signal (varying voltage) by changing electrical signal (varying voltage) by changing dd..

TheThe tuner tuner in a radio is a in a radio is a variable capacitorvariable capacitor. The changing . The changing area area A A alters capacitance until desired signal is obtained.alters capacitance until desired signal is obtained.

Energy of Charged Energy of Charged CapacitorCapacitor

The The potential energypotential energy UU of a of a charged capacitor is equal to charged capacitor is equal to the work (the work (qVqV) required to ) required to charge the capacitor.charge the capacitor.If we consider the average If we consider the average potential difference from 0 to Vpotential difference from 0 to Vf f to be to be V/2V/2::

Work = Q(V/2) = ½QVWork = Q(V/2) = ½QV

221 1

2 2; ; 2

QU QV U CV U

C

221 1

2 2; ; 2

QU QV U CV U

C

Example 3:Example 3: In a capacitor, we found its In a capacitor, we found its capacitance to be capacitance to be 11.1 nF11.1 nF, the voltage , the voltage 200 V200 V, and the charge , and the charge 2.22 2.22 CC. Find the . Find the potential energy potential energy UU..

212 (11.1 nF)(200 V)U

U = 222 JU = 222 J

212U CV

212U CV

Verify your answer from Verify your answer from the other formulas for the other formulas for P.E.P.E.

2

12 ;

2

QU QV U

C

2

12 ;

2

QU QV U

C

C = 11.1 nF

200 V

Q = 2.22 C

U = ?

Capacitor Capacitor of of Example Example 33..

Electrical Circuit SymbolsElectrical Circuit Symbols

Electrical circuitsElectrical circuits often contain two or often contain two or more capacitors grouped together and more capacitors grouped together and attached to an energy source, such as attached to an energy source, such as a battery.a battery.

The following symbols are often The following symbols are often used:used:

+

Capacitor

+--+ - + -

- + - + -

Ground Battery-+

Capacitors in SeriesCapacitors in Series

Capacitors or other devices Capacitors or other devices connected along a single path are connected along a single path are said to be connected in said to be connected in seriesseries. See . See circuit below:circuit below:

Series connection of

capacitors. “+ to – to + …”

Charge inside dots is

induced.

Battery

C1 C2C3

++

--

++

++

--

--

Charge on Capacitors in Charge on Capacitors in SeriesSeries

Since inside charge is only Since inside charge is only inducedinduced, , the the chargecharge on each capacitor is the on each capacitor is the samesame..

Charge is same: series connection of capacitors.

Q = Q1 = Q2 =Q3

Battery

C1 C2C3

++

--

++

++

--

--

Q1 Q2 Q3

Voltage on Capacitors in Voltage on Capacitors in SeriesSeries

Since the Since the potential differencepotential difference between between points points AA and and BB is independent of path, is independent of path, the battery voltage the battery voltage V V must equal the must equal the sum of the voltages across each sum of the voltages across each capacitor.capacitor.

Total voltage V Series

connection Sum of voltages

V = V1 + V2 + V3

Battery

C1 C2C3

++

--

++

++

--

--

V1 V2 V3

• •A B

Equivalent Capacitance: Equivalent Capacitance: SeriesSeries

V = V1 + V2 + V3

Q1= Q2 = Q3

++

--

++

++

--

--

C1 C2 C3

V1 V2 V3 ; Q Q

C VV C

31 2

1 2 3

QQ Q Q

C C C C

1 2 3

1 1 1 1

eC C C C

Equivalent Equivalent CCe e for for capacitors capacitors in series:in series:

1

1 1n

ie iC C

1

1 1n

ie iC C

Example 1.Example 1. Find the equivalent Find the equivalent capacitance of the three capacitors capacitance of the three capacitors connected in series with a 24-V battery.connected in series with a 24-V battery.

++

--

++

++

--

--

2 F

C1 C2 C3

24 V

4 F

6 F

1

1 1n

ie iC C

1

1 1n

ie iC C

CCee for for series:series:

1 1 1 1

2 4 6eC F F F

10.500 0.250 0.167

eC

1 10.917 or

0.917ee

CC

Ce = 1.09 F

Ce = 1.09 F

Example 1 (Cont.):Example 1 (Cont.): The equivalent The equivalent circuit can be shown as follows with circuit can be shown as follows with single Csingle Ce.e.

++

--

++

++

--

--

2 F

C1 C2 C3

24 V

4 F 6 F

1.09 F

Ce

24 V

1

1 1n

ie iC C

1

1 1n

ie iC C

Ce = 1.09 F

Ce = 1.09 F

Note that the equivalent Note that the equivalent capacitance capacitance CCee for capacitors in for capacitors in seriesseries is always is always less than the leastless than the least in the circuit. (1.09 in the circuit. (1.09 F < 2 < 2 F)

1.09 F

Ce

24 V

++

--

++

++

--

--

2 F

C1 C2 C3

24 V

4 F 6 F

QC

V

Q CV

Ce = 1.09 F

Ce = 1.09 F

QQTT = C = CeeV = V = (1.09 (1.09 F)(24 F)(24 V);V);

QT= 26.2C

QT= 26.2C

For series For series circuits: circuits: QQTT = Q = Q11

= Q= Q22 = Q = Q33

Q1 = Q2 = Q3 = 26.2 C

Q1 = Q2 = Q3 = 26.2 C

Example 1 (Cont.):Example 1 (Cont.): What is the total What is the total charge and the charge on each charge and the charge on each capacitor?capacitor?

++

--

++

++

--

--

2 F

C1 C2 C3

24 V

4 F 6 F

; Q Q

C VV C

VT= 24 V

VT= 24 V

11

1

26.2 1

C3.1 V

2 F

QV

C

22

2

26.2 6

C.55 V

4 F

QV

C

33

3

26.2 4

C.37 V

6 F

QV

C

Note: VT = 13.1 V + 6.55 V + 4.37 V = 24.0 V

Note: VT = 13.1 V + 6.55 V + 4.37 V = 24.0 V

Example 1 (Cont.):Example 1 (Cont.): What is the voltage What is the voltage across each capacitor?across each capacitor?

Short Cut: Two Series Short Cut: Two Series CapacitorsCapacitors

The equivalent capacitance The equivalent capacitance CCee for for twotwo series capacitors is the series capacitors is the product divided product divided by the sumby the sum..

1 2

1 1 1;

eC C C 1 2

1 2e

C CC

C C

1 2

1 2e

C CC

C C

3 F 6 F

++

--

++

--

C1 C2

ExamplExample:e:

(3 F)(6 F)

3 F 6 FeC

Ce = 2 F

Ce = 2 F

Parallel CircuitsParallel CircuitsCapacitors which are all connected to Capacitors which are all connected to the same source of potential are said the same source of potential are said to be connected in to be connected in parallelparallel. See . See below:below:

Parallel capacitors: “+ to

+; - to -”C2 C3

C1 ++

--

++

--++

--Charges: QT = Q1 + Q2 +

Q3

Voltages: VT = V1 = V2 =

V3

Equivalent Capacitance: Equivalent Capacitance: ParallelParallel

Q = Q1 + Q2 + Q3

; Q

C Q CVV

Equivalent Equivalent CCe e for for capacitors capacitors in parallel:in parallel:

1

n

e ii

C C

1

n

e ii

C C

Equal Voltages: Equal Voltages: CVCV = C= C11VV11 + C + C22VV22 + +

CC33VV33

Parallel capacitors in

Parallel:C2

C3

C1

++

--

++

--

++

--

CCee = C = C11 + C + C22 + + CC33

Example 2.Example 2. Find the Find the equivalent equivalent capacitancecapacitance of the three capacitors of the three capacitors connected in connected in parallelparallel with a 24-V with a 24-V battery.battery.

CCee for for paralleparallel:l:

Ce = 12 FCe = 12 F

C2C3

C1

2 F 4 F 6 F

24 V

Q = Q1 + Q2 + Q3

VT = V1 = V2 = V3

1

n

e ii

C C

1

n

e ii

C C

CCee = (2 + 4 + 6) = (2 + 4 + 6) FF

Note that the equivalent capacitance Note that the equivalent capacitance CCee for capacitors in for capacitors in parallelparallel is always is always greater than the largestgreater than the largest in the circuit. in the circuit. (12 (12 F > 6 > 6 F)

Example 2 (Cont.)Example 2 (Cont.) Find the Find the totaltotal charge Qcharge QTT and and chargecharge across each across each capacitor.capacitor.

Ce = 12 FCe = 12 F

C2C3

C1

2 F 4 F 6 F

24 V

Q = Q1 + Q2 + Q3

V1 = V2 = V3 = 24 V

; Q

C Q CVV

QQ11 = = (2 (2 F)(24 V) = F)(24 V) = 48 48 CCQQ11 = = (4 (4 F)(24 V) = F)(24 V) = 96 96 CCQQ11 = = (6 (6 F)(24 V) = F)(24 V) = 144 144 CC

QQTT = C = CeeVV

QQTT = (12 = (12 F)(24 F)(24 V) V)

QT = 288 C

QT = 288 C

Example 3. Example 3. Find the equivalent Find the equivalent capacitance of the circuit drawn below.capacitance of the circuit drawn below.

C1

4 F

3 F

6 F

24 V

C2

C3

C1

4 F

2 F24 V C3,6 Ce 6 F

24 V

3,6

(3 F)(6 F)2 F

3 F 6 FC

CCee = 4 = 4 F + 2 F + 2 FF

Ce = 6 F

Ce = 6 F

Example 3 (Cont.)Example 3 (Cont.) Find the total charge Find the total charge QQTT. .

C1

4 F

3 F

6 F

24 V

C2

C3

Ce = 6 F

Ce = 6 F

Q = CVQ = CV = (6 = (6 F)(24 F)(24 V)V)

QT = 144 CQT = 144 C

C1

4 F

2 F24 V C3,6 Ce 6 F

24 V

Example 3 (Cont.)Example 3 (Cont.) Find the charge Find the charge QQ44 and voltage and voltage VV44 across the the 4across the the 4F F capacitorcapacitor

C1

4 F

3 F

6 F

24 V

C2

C3

V4 = VT = 24 V

V4 = VT = 24 V

QQ44 = = (4 (4 F)(24 F)(24 V)V)

Q4 = 96 C

Q4 = 96 C

The remainder of the charge: (144 The remainder of the charge: (144 C – C – 96 96 C) is on C) is on EACH EACH of the other capacitors. of the other capacitors. (Series)(Series)

Q3 = Q6 = 48 C

Q3 = Q6 = 48 C

This can also be found from Q = C3,6V3,6 = (2 F)(24 V)

This can also be found from Q = C3,6V3,6 = (2 F)(24 V)

Example 3 (Cont.)Example 3 (Cont.) Find the Find the voltagesvoltages across the across the 33 and and 6-6-FF capacitors capacitors

C1

4 F

3 F

6 F

24 V

C2

C3

Note: V3 + V6 = 16.0 V + 8.00 V = 24 V

Note: V3 + V6 = 16.0 V + 8.00 V = 24 V

Q3 = Q6 = 48 C

Q3 = Q6 = 48 C

3

48 C16.

3V

F0V

6

48 C8.0

6V

F0V

Use these techniques to find voltage and capacitance across each capacitor in a

circuit.

Use these techniques to find voltage and capacitance across each capacitor in a

circuit.

Summary: Series CircuitsSummary: Series Circuits

1

1 1n

ie iC C

1

1 1n

ie iC C

Q = Q1 = Q2 = Q3

Q = Q1 = Q2 = Q3

V = V1 + V2 + V3

V = V1 + V2 + V3

1 2

1 2e

C CC

C C

1 2

1 2e

C CC

C C

For two capacitors at a For two capacitors at a time:time:

Summary: Parallel CircuitsSummary: Parallel Circuits

Q = Q1 + Q2 + Q3

Q = Q1 + Q2 + Q3

V = V1 = V2 =V3V = V1 = V2 =V31

n

e ii

C C

1

n

e ii

C C

For complex circuits, reduce the circuit in steps using the rules for both series and parallel connections until you are able to solve problem.

For complex circuits, reduce the circuit in steps using the rules for both series and parallel connections until you are able to solve problem.

AP Physics HW 3/26 AP Physics HW 3/26 • Read Chapter 16.3 and 16.5 (that’s it!)Read Chapter 16.3 and 16.5 (that’s it!)

• Do Problems…#63, 65, 67, 69, 71, 93, Do Problems…#63, 65, 67, 69, 71, 93, 95 95

• Finish ALL other HW problems setsFinish ALL other HW problems sets

• Yosemite $ DUE.Yosemite $ DUE.

• T-Shirt Design? ...25 points up for grabs!T-Shirt Design? ...25 points up for grabs!